An approximation property of Pisot numbers

Let q > 1. Initiated by P. Erdos et al. in [4], several authors studied the numbers l(m)(q) = inf{y: y is an element of Lambda(m), y not equal 0}, m = 1, 2,..., where Lambda(m) denotes the set of all finite sums of the form y = epsilon(0) + epsilon(1) q + epsilon(2)q(2) + ... + epsilon(n)q(n) with integer coefficients -m less than or equal to epsilon(i) less than or equal to m. It is known ([1], [4], [6]) that q is a Pisot number if and only if l(m)(q) > 0 for all m. The value of l(1)(q) was determined for many particular Pisot numbers. but the general case remains widely open. In this paper we determine the value of l(m)(q) in other cases. (C) 2000 Academic Press.